Nonnegative normal matrices

Nonnegative normal matrices

Nonnegative Normal Matrices S. K. Jain and L. E. Snyder Department of Mathematics Ohio University Athens, Ohio 45701 Submitted by Richard A. Bruald...

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Nonnegative Normal Matrices S. K. Jain and L. E. Snyder Department

of Mathematics

Ohio University Athens, Ohio 45701

Submitted by Richard A. Brualdi

ABSTRACT A structural for which nonnegative

1.

characterization

the transpose coefficients

is given for the class of those nonnegative

is a polynomial and no constant

in the matrix with the polynomial

matrices having

term.

INTRODUCTION

A large number of characterizations of the important class of normal matrices are given by Grone et al. in [2]. One of the characterizations for a real normal matrix is that its transpose is a polynomial in the matrix. A characterization for the class of those stochastic matrices for which the transpose is equal to some power of the matrix is given by Sir&horn in [4]. Our result extends Sinkhorn’s result to any nonnegative A for which AT = (Y, A”’ + . + ak A”‘k, where each cxi > 0 and 0 < rnl < m2 < *** < mk.

2.

DEFINITIONS

AND

NOTATION

A matrix A = (aij> is said to be nonnegative if aij >, 0 for all i,j, and A is said to be positive if aii > 0 for all i, j. A real matrix A is said to be normal if A commutes with AT, the transpose of A. X is said to be a l-inverse of A if AXA = A, and is said to be a {1,2}-inverse of A if in addition XAX = X. A is said to be stochastic if A is nonnegative and the LINEAR

ALGEBRA

AND ITS APPLICATIONS

0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010

182: 147-155

(1993) 0024-3795/93/$6.00

147

148

S. K. JAIN AND L. E. SNYDER

sum of each row is one. A = (qj> is called a Z-matrix if aij Q 0 for i # j; A is an M-matrix if A can be expressed in the form A = SI - B, where s > 0, B > 0, and s > p(B), the spectral radius of B.

3.

MAIN

RESULTS

THEOREM 1.

If A is a nonnegative,

nonzero

matrix such that

AT = p(A), where p is a polynomial, p(t)

= altm’

+ CQtm + e-0 +a,YQ,

with all czi > 0, and 0 < ml < m2 < or there is a permutation

*** < mk, then either A is symmetric

matrix P such that

where J is a direct sum of matrices of the following (I)

pxxT,

types:

where x is a positive column vector with xTx = 1,

(II) (

0

0

. . .

0

0

*2x3 T

0

...

0

;

;

0

0

0

...

0

xd- 1xd’

XdXT

0

0

...

0

0

0

x1x2

0 p;

T

where the xi’s are positive unit column vectors, possibly of diferent orders, and P is the unique positive fixed point of the polynomial p. For each summand of type (II) which is a d X d block matrix, for each of the exponents m, in the polynomial p. Proof.

By hypothesis AT =p(A),

d must divide mi + 1

NONNEGATIVE where

NORMAL

149

MATRICES

p is a polynomial,

PW

=

such that oi > 0 and mi > 0 for each i. Then A = c~,[[p(A)]‘~ First we determine

the structure

A = p( AT) = p( p( A)), i.e.,

+ . . . +q[p(A)]? of matrices

(1)

satisfying the polynomial

equa-

tion (1) with positive coefficients. Suppose m, = 1. Then from (1) we obtain (1 -

,;)A

= cyi( crsAn,

+ . . .) + a,[ p( A)]-

+ a.. +q[

p( A)]? (2)

If (pi = 1, then c+ = ~ya = a** = ok = 0, since the left side of (2) is the zero matrix and so k = 1. Hence A is symmetric. If oi # 1, then (pi < 1 since the right side of Equation (2) is nonnegative. Equation (2) thus allows us to express

A as A = A’X,

(3)

where X is a nonnegative matrix which commutes with A. Consequently, A has a nonnegative I-inverse. In case m, > 2, it follows easily from Equation (1) that A can be expressed as in (31, so again A has a nonnegative l-inverse. If X is a nonnegative l-inverse of A, then Y = XAX {l, 2}-inverse of A. Consequently, matrix P such that

by Theorem

where C and D are some nonnegative matrices a direct sum of matrices of the following types:

is a nonnegative

1 of [3] there is a permutation

of appropriate

sizes and ] is

150

S. K. JAIN AND L. E. SNYDER (I)

PxyT, where

x and y are positive vectors with yTx = 1,

(II)

I

PlZW2T

0

0

0

‘..

0

0

“.

0

0

0

P23GY3T

0

0

0

Pdl%iYT

0

0

.

.

.

0

.

.

0

Pd-ldXd-1Y: 0

where xi and yi are positive vectors of the same order with yTxi = 1 and the Pij’s are positive. xi and xj are not necessarily of the same order if i #j. Since AT = p(A), it is also true that MT = PATPT = Pp( A)PT = p(PAPT) = p(M). F rom this it follows that the (2,l) block of M T must be zero matrices, or equivalently, That is, M=PAPT=

block and the (1,s) JO = 0 and Cj = 0.

(‘0 ; i

The matrix J also will satisfy the condition that J’ = p(J), direct sum of matrices implies that each of the summands

and J being a S will likewise

satisfy the condition ST = p(S) for the same polynomial p. Let us consider a summand S of type I, i.e., S = PxyT, with yTx = 1 and x, y being positive vectors. Note that (xyT)j = xyT for all j z 1. Hence pyxT

=

Postmultiply

p( pxyT) = ( a1 pm’ +

p”’

a2

+ ***

+a, p-)

xyT.

(4)

both sides of (4) by x, then (PxTx)y

=

(aJF

+

-0.

+cQ~m~)X.

(5)

It follows that y equals a positive scalar times the positive vector x. Consequently there is no loss in generality in assuming that S has the form S = /3xxT, with x being a positive unit vector. Then (4) becomes

which implies that /3 = p( P> so that P is the unique positive fxed the polynomial p.

point of

NONNEGATIVE

NORMAL

151

MATRICES

Next let us consider a summand when d = 4, then

S of the type II and the powers of S. For

example,

0

S=

0

0

P&Y3T

0

PlZXlY2T

0

0

0

0

0

0

0

&X*YT

P34X3Y4T 0

0

0

0

0

0

0

0

0

0

0

PlZ

0

Yl

&3x1

P23P34QY4T

$2 = P34P41~3Y: 0

,

s3

P41 PlZ

Yz’

0 P23P34P41GYT

=

0

\ and S4

x4

is a block

' I

0

0

PlZ P23P34Xl Y4'

0

0

0

0

0

&P41&X3Yi-

0

O

\

P41&2&3X4Y3T

0

i

'

I

diagonal matrix with the ith block haying the form /.Lx,yT,

where I-L= PiZ Pas P34 P4i. I n general Sd is a block diagonal matrix and the ith block is pxi y,r, where p = &&a ... Pdi. Likewise S qd is a block diagonal matrix for any q, and the ith block is /J~x~yT. From this it follows that for 0 < r < d, Sdq+r = ~4s’. Each of the exponents rni in the polynomial p can be expressed as mi = dqi + ri, where 0 < ri < d. Then

ST = p(S)

Since each (Y~> 0 and only corresponding nonzero blocks Therefore, for any d > 2, of the form mi = dqi + d (6) are equal to

=

c

qyQ’+r’

=

c

(yi

/.,&q’s’:.

the nonzero blocks of Sd-i can match the of ST, it follows that each ri = d - 1. the exponents mi of the polynomial p must be 1. The (1, d) blocks in both sides of Equation

S. K. JAIN AND L. E. SNYDER

152

Multiplying this equation on the right by xd leads to the result that yi is a positive scalar multiple of xi. Similarly it can be shown that each yi is a positive scalar multiple of xi, and so without loss of generality it may be assumed that each xi is a unit vector and that yi = xi for each i. Now let us equate the corresponding

blocks

of the

diagonal

in the

equation SST = STS. For the (1, 1) blocks we have p,“, x1x: = P$ixixT, and for the other blocks we have fi,“, x2 xi = p,“, x2 xi, p,Z, x3 XT = p,“, x3 xi, etc. Hence & = p2a = pa4 = .a* = pdl. Thus the matrix S must have the form

0

x1x2’

0

T

0

...

0

0

.

0

0

0

0

0

0

...

0

0

0

...

0

x2

x3

s=p XdXT

Let C = (l/p>S. Since C”: = CT whenever follows from ST = p(S) that P = p( P>.

Xd-

14

0

I

m, + 1 is a multiple

of d, it n

REMARK 1. If xyT is stochastic, then all components of x are equal. Furthermore, if all corn onents of y are also equal though not necessarily the same as for x, then xy K - Jk, where k is the size of x and Jk is the square matrix of order k with all entries equal to l/k. A normal stochastic matrix A is doubly stochastic (see [4, p. 225]>, i.e., AT is stochastic. Sinkhorn’s structural characterization [4] of those stochastic matrices A for which AT = Ak, follows from Theorem 1 and Remark 1. The conditions in Theorem 1 are also sufficient. THEOREM 2. Zf there is a permutation matrix P such that PAPT is a direct sum of the nonnegative matrices of the types (I) and (II), then there is a monomial p with p( P) = p and AT = p(A).

Proof. Note that if S is a is a periodic matrix, i.e., Cd+l each summand of type (II) summands occur, then A will

summand of type (II), then = C. Let m = Icm(d,, d,, . 1s a dj X di block matrix. be symmetric and the result

S = PC, where C . , d,) - 1, where [If only type (I) is obvious.] Then

NONNEGATIVE S”

=

NORMAL

MATRICES

pmcm = pmCT = pmAIST.

then p(S)

= ST for each summand

153

Hence, if we let p(t) = (l/P”‘-‘It”, S; therefore p(A) = AT,

n

REMARK 2. Note that a positive normal matrix A which is not symmetric cannot satisfy the relation p(A) = AT for any polynomial p with nonnegative coefficients

and no constant

term. For example,

the circulant

matrix

is such a matrix. A natural question arises as to what the structure is for matrices A for which AT = p(A) where p(t) is a polynomial with nonnegative coefficients and p(O) # 0. We have not been able to settle this question, but we give below some observations, contained in Propositions 1 and 2, which may be of interest.

PROPOSITION 1. lfA is a nonnegative

AT= p(A), with cq,, CY~,

where

p(A)

matrix such that

= (~~1 + crlAml + . . . okA”‘k,

, CQ positive, then

(i) A has at most two positive eigenvalues, (ii) A-’ exists and is an M-matrix. Proof.

(i): First we note that if A, is a positive eigenvalue

of A, then

p(h,) is also a positive eigenvalue of A because A and AT have the same eigenvalues. Since the function p is strictly increasing on the interval [O, m), it follows that any positive eigenvalue of A must be a fixed point of p. Also, the graph of p is concave upward on the interval [O, m), and so there can be at most two such fmed points. (ii): From AT = p(A), we have

A = p( AT) = p( p(A))

= a,Z + q[

p( A)]-

+ a** +(Y~[ p( A)lmk,

154

S. K. JAIN AND L. E. SNYDER

so A = p(a,)Z + Aq(A) f or some polynomial cients. Note that p(cz,,> > 0, so

I=

exists and that A-’

N% of [l, p. 1371, A-’

is a Z-matrix [since C$ A) > 01. By

is an M-matrix.

PROPOSITION 2. Zf A is a nonsymmetric AT = p(A), where deg p(t) > 2. Proof.

coeffi-

&[A -A@)] = &[Z - q(A)]A.

This shows that A-’ condition

4 with nonnegative

p(t)

has nonnegative

Assume deg p(t)

= 2. From

R matrix

coefficients

with A 2 0 and and

p(O) # 0,

if

then

A = p( p( A)) we have

O=p(p(A))-A=&,Z-&A+&A2+&A3+&A4p

(7)

where

&,, PI, and p4 are positive and the other P’s are nonnegative. From Proposition 1 we know that A -’ is an M-matrix, so the eigenvalues of A-’ and A must have positive real parts (condition F,, of [l, p. 1501 and that A has at most two real eigenvalues. Hence the minimum polynomial of A must contain a factor t - A, and a factor of the form t2 - at + b, where A,, a, and b are positive. The product of these two factors, say (t - AlXt2 - at + b) = t3 - y2 t2 + y,t - yO, with each -yi > 0 and y0 > 0, must divide the right side of (7). The quotient must be of the form P4(t - h2) with A, > 0, since &, > 0. However, this would then contradict the fact that & > 0. W Hence we must have deg p(t) > 2, completing the proof.

EXAMPLE. The following matrix A satisfies the equation + $A2 and has the eigenvalues

A, = 2 -

fi

AT = +Z + iA

and A, = 2 + fi:

Although it is possible to find larger matrices which satisfy the same polynomial equation in the example above, Proposition 2 shows that any such matrix must be symmetric. The authors have been unable to find a nonsymand metric matrix A and a polynomial p(t) with nonnegative coefficients p(O) # 0 for which

AT = p(A).

NONNEGATIVE

NORMAL

The authors

wish

155

MATRICES

to thank

the referee

for

his helpful

suggestions

and

comments. REFERENCES 1

A.

Berman

and

R. J. Plemmons,

Sciences, Academic, 2

R. Grone, Algebra

3

R.

matrices,

Sinkhorn,

40:225-228

Matrices

in the

Mathematical

E. M. Sa, and H. Wolkowicz,

Trans.

Power

Normal

matrices,

Linear

of nonnegative

group-

(1987).

87:213-225

S. K. Jain, E. K. Kwak, and V. K. Goel, monotone

4

C. R. Johnson,

Appl.

Nonnegative

New York, 1979.

Amer.

symmetric

Math.

Decomposition Sot. 257:371-385

stochastic

matrices,

(1980). Linear

(1981).

Received 24 ]uly 1991; final manuscript

accepted 26 January 1992

Algebra

Appl.